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First-order power-source rates in polynomial and exponential spectra

Example

Assume Countable Choice. Let $X$ and $Y$ be real Hilbert spaces, let $K : X \to Y$ be a compact operator of infinite rank, and write $s_{1} \geq s_{2} \geq \dots > 0$ , $s_{j} \to 0$ , for its positive singular values. Set $Q = ρ > 0$ and $ν = 1$ . Over the minimum-norm power source ball $X_{1, ρ} (K)$ in the Gaussian sequence inverse model associated with this singular system, if $s_{j} ≍ j^{- 1}$ then the minimax RMS error is of order $ϵ^{2 / 5}$ ; if $s_{j} ≍ e^{- j}$ then it is of order $ϵ^{1 / 2}$ as $ϵ ↓ 0$ . The constants may depend on $ρ$ and the fixed singular-value comparison constants.

Example references

Power source balls are weighted statistical ellipsoids

Polynomial and exponential minimax estimation rates

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Definition 2 and §3, Table 1

Facts & Assumptions

[F1]

Power source balls are weighted statistical ellipsoids: $X_{ν, ρ} (K)$ has ellipsoid weights $a_{j} = s_{j}^{- ν}$ and radius $Q = ρ$ .

[F2]

Polynomial and exponential minimax estimation rates: For polynomial singular values with polynomial axes, the rate is $ϵ^{κ / (κ + p + 1 / 2)}$ ; for exponential singular values with exponential axes, the rate is $ϵ^{κ / (p + κ)}$ .

[F3]

Power source ball and Gaussian sequence inverse model: The source class is $X_{ν, ρ} (K) = {∣ K ∣^{ν} w : ∥ w ∥_{X} \leq ρ}$ in the real Gaussian sequence inverse model.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is inherited through the compact singular system.

Given: Countable Choice, real Hilbert spaces $X, Y$ , a compact infinite-rank operator $K : X \to Y$ with positive singular values $s_{1} \geq s_{2} \geq \dots > 0$ and $s_{j} \to 0$ , $ρ > 0$ , $ν = 1$ , and either $s_{j} ≍ j^{- 1}$ or $s_{j} ≍ e^{- j}$ .

Verification

1.1givenF1F2algebra

If $s_{j} ≍ j^{- 1}$ , [F1] gives $a_{j} = s_{j}^{- 1} ≍ j$ , so the first formula of [F2] applies with $p = κ = 1$ and gives $ϵ^{2 / 5}$ .

1.2givenF1F2algebra

If $s_{j} ≍ e^{- j}$ , [F1] gives $a_{j} = s_{j}^{- 1} ≍ e^{j}$ , so the second formula of [F2] applies with $p = κ = 1$ and gives $ϵ^{1 / 2}$ .

2.1givenA1F1F2F3step 1.1step 1.2algebra∎

By [A1], the compact singular-system coordinate model is interpreted with its stated Countable Choice prerequisite. Both substitutions use the exact coefficient ellipsoid from [F1] and preserve the stated radius $ρ$ , so they yield the two source-ball rates in the Example.

Depends on

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Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Definition 2 and §3, Table 1, PDF pp. 2 and 5–6 (standard reference, not scraped)